# The Canadian Consumer Price Index Reference Paper

Appendix – Common Price Index Formulae

Common index formulae for elementary price indices (lower level) between previous period t-1 and current period t |
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Name |
Index formulae |
Description |

Dutot | ${I}_{D,a}^{t-1:t}=\frac{{\displaystyle \sum _{i=1}^{n}\frac{1}{n}{p}_{i}^{t}}}{{\displaystyle \sum _{i=1}^{n}\frac{1}{n}{p}_{i}^{t-1}}}$ | A price index defined as the ratio of the unweighted arithmetic average of the prices in the current period t to the unweighted arithmetic average of the prices in period t-1. See chapter 6, formula 6.3. |

Jevons | ${I}_{J,a}^{t-1:t}=\frac{{\displaystyle \prod _{i=1}^{n}{({p}_{i}^{t})}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}}}{{\displaystyle \prod _{i=1}^{n}{({p}_{i}^{t-1})}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}}}$ | A price index defined as the ratio of the unweighted geometric average of the prices in the current period t to the unweighted geometric average of the prices in
period t-1. See chapter 6, formula 6.2. |

Weighted Jevons | ${I}_{WJ,a}^{t-1:t}=\frac{{\displaystyle \prod _{i=1}^{n}{\left({p}_{i}^{t}\right)}^{{w}_{i}/{\displaystyle \sum _{i=1}^{n}{w}_{i}}}}}{{\displaystyle \prod _{i=1}^{n}{\left({p}_{i}^{t-1}\right)}^{{w}_{i}/{\displaystyle \sum _{i=1}^{n}{w}_{i}}}}}$ | A price index defined as the ratio of the explicitly weighted geometric average of the prices in the current period t to the explicitly weighted geometric average of the prices in period t-1. See chapter 6, formula 6.4. |

Common index formulae for aggregate price indices (upper level) between the price reference period 0 and current period
t |
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Name |
Index formulae |
Description |

Fisher | ${I}_{F,A}^{0:t}={\left({I}_{L,A}^{0:t}\times {I}_{P,A}^{0:t}\right)}^{\frac{1}{2}}$ | A price index defined as a geometric average of the Laspeyres price index and the Paasche price index. It is a symmetrically weighted index using quantities of goods and services from both periods 0 and t. |

Laspeyres | ${I}_{L,A}^{0:t}=\frac{{\displaystyle \sum _{i=1}^{n}{p}_{i}^{t}{q}_{i}^{0}}}{{\displaystyle \sum _{i=1}^{n}{p}_{i}^{0}{q}_{i}^{0}}}$ | A price index defined as an asymmetrically weighted fixed-basket index that uses the quantities of goods and services from the base period 0. See chapter 6, formula 6.5. |

Lowe | ${I}_{Lo,A}^{0:t}=\frac{{\displaystyle \sum _{i=1}^{n}{p}_{i}^{t}{q}_{i}^{b}}}{{\displaystyle \sum _{i=1}^{n}{p}_{i}^{0}{q}_{i}^{b}}}$ | A price index defined as an asymmetrically weighted fixed-basket index that uses the quantities of goods and services from the chosen weight reference period b. See chapter 6, formula 6.6. |

Marshall-Edgeworth | ${I}_{ME,A}^{0:t}=\frac{{\displaystyle \sum _{i=1}^{n}{p}_{i}^{t}\times \left[\frac{\left({q}_{i}^{0}+{q}_{i}^{t}\right)}{2}\right]}}{{\displaystyle \sum _{i=1}^{n}{p}_{i}^{0}\times \left[\frac{\left({q}_{i}^{0}+{q}_{i}^{t}\right)}{2}\right]}}$ | A price index defined as the ratio of average weighted prices between period 0 and t with weights as the arithmetic average of quantities from both periods 0 and t. It is a symmetrically weighted fixed-basket index. |

Paasche | ${I}_{P,A}^{0:t}=\frac{{\displaystyle \sum _{i=1}^{n}{p}_{i}^{t}{q}_{i}^{t}}}{{\displaystyle \sum _{i=1}^{n}{p}_{i}^{0}{q}_{i}^{t}}}$ | A price index defined as an asymmetrically weighted fixed-basket index that uses the quantities of goods and services from the current period t. |

Törnqvist-Theil | ${I}_{T,A}^{0:t}={\displaystyle \prod _{i=1}^{n}{\left(\frac{{p}_{i}^{t}}{{p}_{i}^{0}}\right)}^{\frac{1}{2}\left({s}_{i}^{0}+{s}_{i}^{t}\right)}}$
${s}_{i}^{t}\equiv \frac{{p}_{i}^{t}{q}_{i}^{t}}{{\displaystyle \sum _{i=1}^{n}{p}_{i}^{t}{q}_{i}^{t}}}$ |
A price index defined as a geometric average of price relatives weighted by the average expenditure shares in both periods 0 and t. It is a symmetrically weighted index. |

Walsh | ${I}_{W,A}^{0:t}=\frac{{\displaystyle \sum _{i=1}^{n}{p}_{i}^{t}\sqrt{{q}_{i}^{t}{q}_{i}^{0}}}}{{\displaystyle \sum _{i=1}^{n}{p}_{i}^{0}\sqrt{{q}_{i}^{t}{q}_{i}^{0}}}}$ | A price index defined as the ratio of average weighted prices between period 0 and t with weights as the geometric average of quantities from both periods 0 and t. It is a symmetrically weighted fixed-basket index. |

Source: Statistics Canada, Consumer Prices Division. |

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